Asymptotic Distribution of Eigenvalues for Damped String Equation: Numerical Approach

In the present paper, we consider a one-parameter family of the nonself-adjoint operators, which are the dynamics generators for systems governed by the wave equations containing dissipative terms. The equations contain viscous damping terms and are equipped with the boundary conditions involving an arbitrary complex parameter. In the current engineering literature, this type of boundary condition is used to model the action of smart materials (self-sensing/self-straining actuators). In the previous research of the first writer, the aforementioned dynamics generators have been studied analytically and precise asymptotic formulas for the eigenvalues have been derived (the asymptotic when the number of the eigenvalues tends to infinity). The goal of the present paper is to demonstrate that the analytic formulas are not only important theoretically, but also extremely efficient practically. Namely, we show that the leading terms in the asymptotic formulas approximate the actual eigenvalues with excellent accuracy. To justify the results, we use two methods, i.e., the Newton method and the Tchebychev method. First, Newton’s method is applied to the characteristic equation using asymptotic formulas as initial guesses to find the eigenvalues. The convergence of Newton’s method is improved by modifying the asymptotic formula. Second, we use Tchebychev discretization to circumvent the nonlinear characteristic equation and to obtain a finite-dimensional generalized eigenvalue problem that approximates the infinite-dimensional one. Finally, to solve the generalized eigenvalue problem, we use the QT algorithm.